There are various types of bevel gears, these types being differentiated, inter alia, on the basis of the profile of the longitudinal flank lines. The following bevel gears are differentiated according to the profile of the longitudinal flank lines:                straight-toothed bevel gears        helical-toothed bevel gears        spiral-toothed bevel gears.        
Bevel gear pair teeth can be uniquely established by the associated virtual plane gear teeth (if the pitch cone angles are known). The corresponding plane gear can also be imagined as a wafer-thin profile disc. The plane gear results from the bevel gear teeth in that the pitch cone angle is set to δp=90°. In general, every perpendicular section through a plane gear tooth has linear flanks. Bevel gears of a bevel gear pair have the same plane gear tooth count.
The term “spiral-toothed bevel gears” is presumed to have been taken from the American world, where these bevel gears are generally referred to as “spiral bevel gears.” It would be better to use the term “arc-toothed bevel gears” instead of “spiral-toothed bevel gears” here, since the spiral already represents a special form of a curve (e.g., the Archimedean). However, the title “spiral-toothed bevel gears” is still used hereafter, as has established itself in practice.
In the case of spiral-toothed bevel gears, a further subdivision is possible with respect to the shape of the longitudinal flank line:                circular arcs        epicycloids, in particular extended epicycloids        evolvents        hypocycloids, in particular extended hypocycloids.        
Circular-arc-toothed bevel gears have a circular arc as a longitudinal flank line. Circular-arc-toothed bevel gears are manufactured in the single indexing method (also referred to as intermittent indexing process, single indexing process, or face milling). The single indexing method is schematically shown in FIG. 1A. The cutter 21 of a cutter head 20 completes a circular movement while the bevel gear 11 to be produced rests in a fixed position. To manufacture further tooth gaps, the cutter head 20 is retracted and the workpiece 11 is rotated by an indexing angle. The step-by-step further rotation (counterclockwise here) is indicated in FIG. 1A by the arrows A, B, and C. Therefore, one tooth gap 22 is always manufactured at a time.
Epicycloidal, in particular expanded-epicycloidal (also referred to as extended-epicycloidal) toothed gearwheels are manufactured by a continuous indexing method (also referred to as continuous hobbing, continuous indexing process, or face hobbing). In the production of the epicycloids in the continuous indexing method, the ratio of plane gear tooth count zp of the bevel gear to number of passes Gx of the bar cutter head (number of the cutter groups) corresponds to the ratio of the radius RG of the base circle GK and the radius RR of the pitch circle RK. One refers to an extended epicycloid when the cutter head nominal radius rc, on which the blades of the cutter 23 are seated, is greater than the radius RR of the pitch circle RK (see FIG. 1B). In this continuous indexing method, both the cutter head and also the workpiece 11 rotate in a movement sequence that is chronologically adapted to one another. The indexing is thus performed continuously and gaps 12 and the corresponding teeth 13 are generated quasi-simultaneously. A corresponding example is shown in FIG. 1B. The cutter head rotates counterclockwise here, while the workpiece 11 rotates clockwise (this rotational movement is also referred to as plane gear rotation). An epicycloid (e.g., an extended epicycloid) is produced here. The movements thus occur here in opposite directions. If both rotate in the same direction, a hypocycloid is produced (as shown in FIG. 1C). FIG. 1B shows that the cutters 23 of a corresponding cutter head are typically situated in pairs (also referred to as in groups in the case of two, three, or more cutters per cutter group). FIG. 1B shows that the pitch circle RK of the cutter head rolls along the base circle GK of the workpiece 11. M refers to the center point of the cutter head here. This center point M is coincident with the center point of the pitch circle RK. The coupling of the two rotational movements is performed so that only one cutter pair, or one cutter group, moves through a tooth gap 12 in each case.
A bevel gear having a “linear” hypocycloid as a longitudinal flank line can be produced according to the principle shown in FIG. 2. The mathematical principle shown is known from various textbooks, but also from European Patent Application EP 1348509 A2. Fundamentally, this approach is also known from “Stanki dja obrabotki konitscheskich zubtschatych kolos,” Izdanie 2-e, V. N. Kedrinskij, K. M. Pismanik, Izdatelstvo “Maschinostroenie” Moskva 1967, pages 506-508.
In order to obtain a hypocycloid, the pitch circle RK having the radius RR rolls in the interior of the fixed base circle GK having radius RG. The pitch circle RK rotates around its axis (center point M), as indicated by the arrow P1. As indicated by the arrow P2, the pitch circle RK rolls counterclockwise in the interior of the base circle GK (the rotational direction could also be reversed). The pointer Z1 is oriented radially outward fixed in place in the pitch circle RK and is associated with a generating point U on the circumference of the pitch circle RK. This point U is fixed in place in the coordinate system of the pitch circle RK, i.e., this point U is fixedly connected to the pitch circle RK. Through the rolling movement of the point U, i.e., through its own rotation around the point M coupled to the satellite movement around the center point of the base circle GK, the point U generates a hypocycloid HY in the x-y coordinate system of the base circle GK, or a straight line in the special case shown. The point U thus defines or describes a linear hypocycloid (HY), when the pitch circle RK rolls in the base circle GK. The cutter head radius rc is rc=RR here. The two circles RK and GK are shown in a Cartesian x-y coordinate system here.
The parameter representation in this x-y coordinate system reads as follows:
                    x        =                                            (                              RG                -                RR                            )                        ⁢            cos            ⁢                                                  ⁢            λ                    -                      RR            ⁢                                                  ⁢            cos            ⁢                                                  ⁢                                          RG                -                RR                            RR                        ⁢            λ                                              (        1        )                                y        =                                            (                              RG                -                RR                            )                        ⁢            sin            ⁢                                                  ⁢            λ                    -                      RR            ⁢                                                  ⁢            sin            ⁢                                                  ⁢                                          RG                -                RR                            RR                        ⁢            λ                                              (        2        )            
In equations (1) and (2), λ represents the rotational angle of the center point M of the pitch circle RK in relation to the center point MG of the base circle GK. A snapshot is shown in FIG. 2 where the following applies: λ=0, x=RG, y=0. The coordinates of U are [RG, 0].
FIGS. 3A-3H show, on the basis of a sequence, that there is a special case in which the hypocycloid HY becomes a straight line. In these figures, the references and reference numerals were intentionally omitted, so as not to impair the clarity of the illustration. However, all features of FIG. 3A-3H that match the features of FIG. 2 are in accordance therewith. A straight line results when the condition RR=RG/2, or RG/RR=2, is met. It may be seen on the basis of FIGS. 3A-3H that the generating point U is displaced starting from the illustration in FIG. 3A (λ=0°) along the x-axis from the coordinate position [RG, 0] to the left to the coordinate position [−RG, 0]. In FIG. 3E (λ=180°), the coordinate position [−RG, 0] is reached. The pitch circle RK now rolls through the two lower quadrants of the x-y coordinate system and the point U moves from the coordinate position [−RG, 0] back to the coordinate position [RG, 0]. The straight line HY is a distance which extends in the figures along the x-axis from [RG, 0] to [−RG, 0].
Special shapes of hypocycloids can also be explained on the basis of the figure shown. The special shapes are generated as follows. If the generating point U is inside or outside the pitch circle RK, one refers to either an abbreviated hypocycloid or correspondingly to an extended hypocycloid. The distance between the center M (see FIG. 2) of the pitch circle RK and the position of the generating point U is described by the parameter c. Therefore, c<RR generates an abbreviated hypocycloid and c>RR generates an extended hypocycloid. An extended hypocycloid having c=1.5 RR is shown in FIG. 4A. The pointer Z2 thus has the length c=1.5 RR (the variable c corresponds to the cutter head nominal radius rc in FIG. 1B). The cutter head nominal radius rc is thus rc=1.5 RR here. An abbreviated hypocycloid having c=0.5 RR is shown in FIG. 4B. The pointer Z3 thus has the length c=0.5 RR. The cutter head nominal radius rc is thus rc=0.5 RR here. In each case an ellipse is generated as the hypocycloid HY, whose parameter representation as a function of the angle λ in the x-y coordinate system reads as follows:
                    x        =                              (                                          RG                2                            +              c                        )                    ⁢                      cos            ⁡                          (                              λ                2                            )                                                          (        3        )                                y        =                              (                                          RG                2                            -              c                        )                    ⁢                      sin            ⁡                          (                              λ                2                            )                                                          (        4        )            If c=RR and RR=RG/2, the linear hypocycloid is obtained as a special case, as already described.
In the face cutter heads, which are used to produce bevel gears, one differentiates between so-called bar cutter heads and profile cutter heads. A bar cutter head is equipped with a large number of bar cutters (e.g., forty), each bar cutter having a shaft and a head area. The head area can be given a desired shape and position by grinding the bar cutter. Bar cutter heads are more productive than profile cutter heads, which contain fewer cutters, and the bar cutters can be re-profiled. In contrast, a profile cutter head is equipped with relief-ground cutters. These die cutters (also referred to as profile cutters) maintain their profile shape on the machining surface upon re-grinding. It is an advantage of bevel gear milling using profile cutters that no special grinding machine is required for the re-grinding of these die cutters. The known Zyklo-Palloid® method, for example, uses such profile cutters to produce spiral bevel gears.
The present invention is concerned with the milling of bevel gears having hypocycloidal teeth and in particular the milling of straight-toothed bevel gears.
The methods currently used for milling straight-toothed bevel gears are hobbing (names of known hobbing methods include: Coniflex®, Konvoid, and Sferoid™) and broaching (also known as the Revacycle® method). Two disc-shaped cutter heads of equal size are used in the case of hobbing, in which the cutters on the outer circumference point radially outward. The axes of the two cutter heads are inclined to one another, so that at the narrowest point the cutters of one cutter head may engage between the cutters of the other. One cutter head is thus used for the left flanks and one cutter head is used for the right flanks. This hobbing of straight-toothed bevel gears is a single indexing method, in which crown gear and bevel gear pinions are hobbed. The broaching method is also a single indexing method in which, the tooth flanks of the crown gear and bevel gear pinions are not generated by envelope cuts as in hobbing. Rather, the cutter profiles in broaching exactly correspond to the shape of the final gap profile of the bevel gear. The broaching method is more productive than hobbing in the single indexing method, but has the disadvantage that a special disc-shaped broaching cutter head having a plurality of various die cutters on the circumference is required for almost every bevel gear (transmission ratio).
These examples prove that the tool expenditure can be too large and costly for many users to be able to produce various bevel gears having hypocycloidal teeth and in particular various straight-toothed bevel gears.